with the gauge parameter \(\Lambda _{\alpha (n-1)}\) 
being real but otherwise unconstrained.
In accordance with [1]}, the super-Cotton tensor \(\mathfrak {W}_{\alpha (n)}(H)\)  associated with \(H_{\alpha (n)}\) 
is defined to satisfy the following three conditions:
(i) it is a primary descendent of \(H_{\alpha (n)}\) ; (ii) it is transverse (or covariantly conserved),
\(0=\mathcal {D}^{\beta }\mathfrak {W}_{\beta \alpha (n-1)}(H) ~;\) 
